The family \(H_{R,\theta }^\sharp \)  has three bulk bands, see (REF ), that we denote by \(+,\,-\)  and 0. The definition above is more general than Definition REF  and works beyond compact parameters, see [1]}. However, the parameter \(\theta \)  is compact here and since the edge mode branches are continuous, the two definitions coincide in our case. One has
\(n_\mathrm {b}^+(R) = n(\mathcal {C}_R,\mu ) = n_\mathrm {a}^0(R).\) 
